2014, 1(2): 299-330. doi: 10.3934/jdg.2014.1.299

Turnpike properties of approximate solutions of dynamic discrete time zero-sum games

1. 

Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel

Received  August 2013 Revised  December 2013 Published  March 2014

We study existence and turnpike properties of approximate solutions for a class of dynamic discrete-time two-player zero-sum games without using convexity-concavity assumptions. We describe the structure of approximate solutions which is independent of the length of the interval, for all sufficiently large intervals and show that approximate solutions are determined mainly by the objective function, and are essentially independent of the choice of interval and endpoint conditions.
Citation: Alexander J. Zaslavski. Turnpike properties of approximate solutions of dynamic discrete time zero-sum games. Journal of Dynamics & Games, 2014, 1 (2) : 299-330. doi: 10.3934/jdg.2014.1.299
References:
[1]

O. Alvarez and M. Bardi, Ergodic problems in differential games,, in Advances in Dynamic Game Theory, 9 (2007), 131. doi: 10.1007/978-0-8176-4553-3_7.

[2]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Wiley Interscience, (1984).

[3]

S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions I,, Physica D, 8 (1983), 381. doi: 10.1016/0167-2789(83)90233-6.

[4]

M. Bardi, On differential games with long-time-average cost,, in Advances in Dynamic Games and their Applications, 10 (2009), 3.

[5]

J. Blot and P. Cartigny, Optimality in infinite-horizon variational problems under sign conditions,, J. Optim. Theory Appl., 106 (2000), 411. doi: 10.1023/A:1004611816252.

[6]

J. Blot and N. Hayek, Sufficient conditions for infinite-horizon calculus of variations problems,, ESAIM Control Optim. Calc. Var., 5 (2000), 279. doi: 10.1051/cocv:2000111.

[7]

V. Gaitsgory and M. Quincampoix, Linear programming approach to deterministic infinite horizon optimal control problems with discounting,, SIAM J. Control and Optimization, 48 (2009), 2480. doi: 10.1137/070696209.

[8]

M. K. Ghosh and K. S. Mallikarjuna Rao, Differential games with ergodic payoff,, SIAM J. Control Optim., 43 (2005), 2020. doi: 10.1137/S0363012903404511.

[9]

H. Jasso-Fuentes and O. Hernandez-Lerma, Characterizations of overtaking optimality for controlled diffusion processes,, Appl. Math. Optim., 57 (2008), 349. doi: 10.1007/s00245-007-9025-6.

[10]

O. Hernandez-Lerma and J. B. Lasserre, Zero-sum stochastic games in Borel spaces: Average payoff criteria,, SIAM J. Control Optim., 39 (2000), 1520. doi: 10.1137/S0363012999361962.

[11]

V. Kolokoltsov and W. Yang, The turnpike theorems for Markov games,, Dynamic Games and Applications, 2 (2012), 294. doi: 10.1007/s13235-012-0047-6.

[12]

A. Leizarowitz and V. J. Mizel, One dimensional infinite horizon variational problems arising in continuum mechanics,, Arch. Rational Mech. Anal., 106 (1989), 161. doi: 10.1007/BF00251430.

[13]

V. Lykina, S. Pickenhain and M. Wagner, Different interpretations of the improper integral objective in an infinite horizon control problem,, J. Math. Anal. Appl, 340 (2008), 498. doi: 10.1016/j.jmaa.2007.08.008.

[14]

M. Marcus and A. J. Zaslavski, The structure of extremals of a class of second order variational problems,, Ann. Inst. H. Poincare, 16 (1999), 593. doi: 10.1016/S0294-1449(99)80029-8.

[15]

L. W. McKenzie, Turnpike theory,, Econometrica, 44 (1976), 841. doi: 10.2307/1911532.

[16]

B. S. Mordukhovich, Optimal control and feedback design of state-constrained parabolic systems in uncertainly conditions,, Appl. Analysis, 90 (2011), 1075. doi: 10.1080/00036811003735840.

[17]

E. Ocana Anaya, P. Cartigny and P. Loisel, Singular infinite horizon calculus of variations. Applications to fisheries management,, J. Nonlinear Convex Anal., 10 (2009), 157.

[18]

S. Pickenhain, V. Lykina and M. Wagner, On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems,, Control Cybernet, 37 (2008), 451.

[19]

T. Prieto-Rumeau and O. Hernandez-Lerma, Bias and overtaking equilibria for zero-sum continuous-time Markov games,, Math. Methods Oper. Res., 61 (2005), 437. doi: 10.1007/s001860400392.

[20]

P. A. Samuelson, A catenary turnpike theorem involving consumption and the golden rule,, American Economic Review, 55 (1965), 486.

[21]

A. J. Zaslavski, Turnpike property for dynamic discrete time zero-sum games,, Abstract and Applied Analysis, 4 (1999), 21. doi: 10.1155/S1085337599000020.

[22]

A. J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control,, Springer, (2006).

[23]

A. J. Zaslavski, The existence and structure of approximate solutions of dynamic discrete time zero-sum games,, Journal of Nonlinear and Convex Analysis, 12 (2011), 49.

[24]

A. J. Zaslavski, Structure of Solutions of Variational Problems,, SpringerBriefs in Optimization, (2013). doi: 10.1007/978-1-4614-6387-0.

[25]

A. J. Zaslavski and A. Leizarowitz, Optimal solutions of linear control systems with nonperiodic integrands,, Mathematics of Operations Research, 22 (1997), 726. doi: 10.1287/moor.22.3.726.

show all references

References:
[1]

O. Alvarez and M. Bardi, Ergodic problems in differential games,, in Advances in Dynamic Game Theory, 9 (2007), 131. doi: 10.1007/978-0-8176-4553-3_7.

[2]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Wiley Interscience, (1984).

[3]

S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions I,, Physica D, 8 (1983), 381. doi: 10.1016/0167-2789(83)90233-6.

[4]

M. Bardi, On differential games with long-time-average cost,, in Advances in Dynamic Games and their Applications, 10 (2009), 3.

[5]

J. Blot and P. Cartigny, Optimality in infinite-horizon variational problems under sign conditions,, J. Optim. Theory Appl., 106 (2000), 411. doi: 10.1023/A:1004611816252.

[6]

J. Blot and N. Hayek, Sufficient conditions for infinite-horizon calculus of variations problems,, ESAIM Control Optim. Calc. Var., 5 (2000), 279. doi: 10.1051/cocv:2000111.

[7]

V. Gaitsgory and M. Quincampoix, Linear programming approach to deterministic infinite horizon optimal control problems with discounting,, SIAM J. Control and Optimization, 48 (2009), 2480. doi: 10.1137/070696209.

[8]

M. K. Ghosh and K. S. Mallikarjuna Rao, Differential games with ergodic payoff,, SIAM J. Control Optim., 43 (2005), 2020. doi: 10.1137/S0363012903404511.

[9]

H. Jasso-Fuentes and O. Hernandez-Lerma, Characterizations of overtaking optimality for controlled diffusion processes,, Appl. Math. Optim., 57 (2008), 349. doi: 10.1007/s00245-007-9025-6.

[10]

O. Hernandez-Lerma and J. B. Lasserre, Zero-sum stochastic games in Borel spaces: Average payoff criteria,, SIAM J. Control Optim., 39 (2000), 1520. doi: 10.1137/S0363012999361962.

[11]

V. Kolokoltsov and W. Yang, The turnpike theorems for Markov games,, Dynamic Games and Applications, 2 (2012), 294. doi: 10.1007/s13235-012-0047-6.

[12]

A. Leizarowitz and V. J. Mizel, One dimensional infinite horizon variational problems arising in continuum mechanics,, Arch. Rational Mech. Anal., 106 (1989), 161. doi: 10.1007/BF00251430.

[13]

V. Lykina, S. Pickenhain and M. Wagner, Different interpretations of the improper integral objective in an infinite horizon control problem,, J. Math. Anal. Appl, 340 (2008), 498. doi: 10.1016/j.jmaa.2007.08.008.

[14]

M. Marcus and A. J. Zaslavski, The structure of extremals of a class of second order variational problems,, Ann. Inst. H. Poincare, 16 (1999), 593. doi: 10.1016/S0294-1449(99)80029-8.

[15]

L. W. McKenzie, Turnpike theory,, Econometrica, 44 (1976), 841. doi: 10.2307/1911532.

[16]

B. S. Mordukhovich, Optimal control and feedback design of state-constrained parabolic systems in uncertainly conditions,, Appl. Analysis, 90 (2011), 1075. doi: 10.1080/00036811003735840.

[17]

E. Ocana Anaya, P. Cartigny and P. Loisel, Singular infinite horizon calculus of variations. Applications to fisheries management,, J. Nonlinear Convex Anal., 10 (2009), 157.

[18]

S. Pickenhain, V. Lykina and M. Wagner, On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems,, Control Cybernet, 37 (2008), 451.

[19]

T. Prieto-Rumeau and O. Hernandez-Lerma, Bias and overtaking equilibria for zero-sum continuous-time Markov games,, Math. Methods Oper. Res., 61 (2005), 437. doi: 10.1007/s001860400392.

[20]

P. A. Samuelson, A catenary turnpike theorem involving consumption and the golden rule,, American Economic Review, 55 (1965), 486.

[21]

A. J. Zaslavski, Turnpike property for dynamic discrete time zero-sum games,, Abstract and Applied Analysis, 4 (1999), 21. doi: 10.1155/S1085337599000020.

[22]

A. J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control,, Springer, (2006).

[23]

A. J. Zaslavski, The existence and structure of approximate solutions of dynamic discrete time zero-sum games,, Journal of Nonlinear and Convex Analysis, 12 (2011), 49.

[24]

A. J. Zaslavski, Structure of Solutions of Variational Problems,, SpringerBriefs in Optimization, (2013). doi: 10.1007/978-1-4614-6387-0.

[25]

A. J. Zaslavski and A. Leizarowitz, Optimal solutions of linear control systems with nonperiodic integrands,, Mathematics of Operations Research, 22 (1997), 726. doi: 10.1287/moor.22.3.726.

[1]

Alexander J. Zaslavski. Structure of approximate solutions of dynamic continuous time zero-sum games. Journal of Dynamics & Games, 2014, 1 (1) : 153-179. doi: 10.3934/jdg.2014.1.153

[2]

Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001

[3]

Marianne Akian, Stéphane Gaubert, Antoine Hochart. Ergodicity conditions for zero-sum games. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3901-3931. doi: 10.3934/dcds.2015.35.3901

[4]

Xiangxiang Huang, Xianping Guo, Jianping Peng. A probability criterion for zero-sum stochastic games. Journal of Dynamics & Games, 2017, 4 (4) : 369-383. doi: 10.3934/jdg.2017020

[5]

Tao Li, Suresh P. Sethi. A review of dynamic Stackelberg game models. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 125-159. doi: 10.3934/dcdsb.2017007

[6]

Zhi-Wei Sun. Unification of zero-sum problems, subset sums and covers of Z. Electronic Research Announcements, 2003, 9: 51-60.

[7]

Beatris A. Escobedo-Trujillo. Discount-sensitive equilibria in zero-sum stochastic differential games. Journal of Dynamics & Games, 2016, 3 (1) : 25-50. doi: 10.3934/jdg.2016002

[8]

Sylvain Sorin, Guillaume Vigeral. Reversibility and oscillations in zero-sum discounted stochastic games. Journal of Dynamics & Games, 2015, 2 (1) : 103-115. doi: 10.3934/jdg.2015.2.103

[9]

Ido Polak, Nicolas Privault. A stochastic newsvendor game with dynamic retail prices. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1-12. doi: 10.3934/jimo.2017072

[10]

Eduardo Espinosa-Avila, Pablo Padilla Longoria, Francisco Hernández-Quiroz. Game theory and dynamic programming in alternate games. Journal of Dynamics & Games, 2017, 4 (3) : 205-216. doi: 10.3934/jdg.2017013

[11]

Fernando Luque-Vásquez, J. Adolfo Minjárez-Sosa. Average optimal strategies for zero-sum Markov games with poorly known payoff function on one side. Journal of Dynamics & Games, 2014, 1 (1) : 105-119. doi: 10.3934/jdg.2014.1.105

[12]

Libin Mou, Jiongmin Yong. Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method. Journal of Industrial & Management Optimization, 2006, 2 (1) : 95-117. doi: 10.3934/jimo.2006.2.95

[13]

Fabien Gensbittel, Miquel Oliu-Barton, Xavier Venel. Existence of the uniform value in zero-sum repeated games with a more informed controller. Journal of Dynamics & Games, 2014, 1 (3) : 411-445. doi: 10.3934/jdg.2014.1.411

[14]

Georgios Konstantinidis. A game theoretic analysis of the cops and robber game. Journal of Dynamics & Games, 2014, 1 (4) : 599-619. doi: 10.3934/jdg.2014.1.599

[15]

Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics & Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537

[16]

Jiahua Zhang, Shu-Cherng Fang, Yifan Xu, Ziteng Wang. A cooperative game with envy. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2049-2066. doi: 10.3934/jimo.2017031

[17]

Lars Grüne, Manuela Sigurani. Numerical event-based ISS controller design via a dynamic game approach. Journal of Computational Dynamics, 2015, 2 (1) : 65-81. doi: 10.3934/jcd.2015.2.65

[18]

Songtao Sun, Qiuhua Zhang, Ryan Loxton, Bin Li. Numerical solution of a pursuit-evasion differential game involving two spacecraft in low earth orbit. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1127-1147. doi: 10.3934/jimo.2015.11.1127

[19]

Alexander J. Zaslavski. The turnpike property of discrete-time control problems arising in economic dynamics. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 861-880. doi: 10.3934/dcdsb.2005.5.861

[20]

Zhenbo Wang, Wenxun Xing, Shu-Cherng Fang. Two-person knapsack game. Journal of Industrial & Management Optimization, 2010, 6 (4) : 847-860. doi: 10.3934/jimo.2010.6.847

 Impact Factor: 

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]